A New Expectation Identity and Its Application in the Calculations of Predictive Powers Assuming Normality

For calculating the predictive powers, we suggest an elegant expectation identity to directly calculate the expectations. We calculate the predictive powers of the hypotheses with a nonzero threshold for five different categories, which are non-sequential trials with classical power and Bayesian power, and sequential trials with hybrid predictions, Bayesian predictions, and classical predictions. Moreover, the calculations of the various predictive powers are illustrated through three examples. Finally, when calculating the average success probability in \ncite{9}, it is tricky to find the predictive distribution for the predictive power, whereas, it is straightforward to utilize the expectation identity for the calculation.